Question: $78$ people attended a baseball game. Everyone there was a fan of either the home team or the away team. The number of home team fans was $21$ less than $2$ times the number of away team fans. How many home team and away team fans attended the game?
Solution: Let $x$ equal the number of home team fans and $y$ equal the number of away team fans. The system of equations is then: ${x+y = 78}$ ${x = 2y-21}$ Solve for $x$ and $y$ using substitution. Since $x$ has already been solved for, substitute ${2y-21}$ for $x$ in the first equation. ${(2y-21)}{+ y = 78}$ Simplify and solve for $y$ $ 2y-21 + y = 78 $ $ 3y-21 = 78 $ $ 3y = 99 $ $ y = \dfrac{99}{3} $ ${y = 33}$ Now that you know ${y = 33}$ , plug it back into ${x = 2y-21}$ to find $x$ ${x = 2}{(33)}{ - 21}$ $x = 66 - 21$ ${x = 45}$ You can also plug ${y = 33}$ into ${x+y = 78}$ and get the same answer for $x$ ${x + }{(33)}{= 78}$ ${x = 45}$ There were $45$ home team fans and $33$ away team fans.